User blog:Wythagoras/Dollar function: final version
Inspired by Hyp cos I decided to remake everything exepted for Bracket Notation. Which rule should you use? #If there is nothing after the $, use rule 1 #If there are any non-nested non-subscript numbers, use rule 2 #If there are any non-nested non-subscript 0's, use rule 3 #If there are any non-nested non-subscript b's, use rule 4 #If the previous things doesn't apply but the lowest level bracket can be solved with normal bracket notation: ##Search in the bracket for the least nested lowest level bracket or number ###If it is a 0: ####If the zero is the only content, use rule 3 ####Otherwise, use rule 5 ###If it is another number, use rule 4 ###If it is a bracket: Return to step 5 #If the lowest level bracket can be solved with extended bracket notation: ##Is the number in the typed bracket a 0, use rule 6 and the subrule if needed ##Otherwise, use rule 4 \(\bullet\) can be anything \(\circ\) is a group of brackets \(\diamond\) is a group of zeroes Extended Bracket Notation This works now a bit like the Buchholz hydra, and the limit is \(\psi(\psi_I(0))\) 1. If there is nothing after the $, the array is solved. The value of the array is the number before the $. 2. \(a\$b\bullet=(a+b)\$\bullet\) 3. \(a\$\circ0\bullet\circ=a\$\circ a\bullet\circ\) 4. \(a\$\circ\bullet+1_c\bullet\circ=a\$\circ\bullet_c\bullet_c...\bullet_c\bullet_c\bullet\circ\) with a \(\bullet\)'s 5. If the bracket contains a zero and the bracket has other content, you can remove the zero. 6. If the active bracket has level k and a zero in it, search for the least nested bracket with level (k-1) with the same array in it, nest that bracket a times in the place of the level k bracket. S1: The outermost bracket is always level 1 S2: If there is no bracket with level (k-1), add it directly after the level k bracket. Analysis \([0_2]\) has level \(\varepsilon_0\) \([10_2]\) has level \(\varepsilon_0\omega\) \([[0_2]0_2]\) has level \(\varepsilon_0^2\) \([[10_2]0_2]\) has level \(\varepsilon_0^\omega\) \([0_20_2]\) has level \(\varepsilon_1\) \([1_2]\) has level \(\varepsilon_\omega\) \([0_2]\) has level \(\varepsilon_{\omega^2}\) \([[[0_2]]_2]\) has level \(\varepsilon_{\varepsilon_0}\) \([[[[[0_2]]_2]]_2]\) has level \(\varepsilon_{\varepsilon_{\varepsilon_0}}\) \([[0_2]_2]\) has level \(\zeta_0\) \([1[0_2]_2]\) has level \(\zeta_0\omega\) \([[0_2]_2[1[0_2]_2]]\) has level \(\zeta_0^\omega\) \([[0_2]_20_2]\) has level \(\varepsilon_{\zeta_0+1}\) \([[0_2]_21_2]\) has level \(\varepsilon_{\zeta_0+\omega}\) \([[0_2]_2[[[0_2]_2]]_2]\) has level \(\varepsilon_{\zeta_02}\) \([[0_2]_2[0_2]_2]\) has level \(\zeta_1\) \([[10_2]_2]\) has level \(\zeta_\omega\) \([[[0_2]0_2]_2]\) has level \(\zeta_{\varepsilon_0}\) \([[[[0_2]_2]0_2]_2]\) has level \(\zeta_{\zeta_0}\) \([[0_20_2]_2]\) has level \(\eta_0\) \([[1_2]_2]\) has level \(\varphi(\omega,0)\) \([[0_2]_2]\) has level \(\varphi(\omega^2,0)\) \([[[[0_2]]_2]_2]\) has level \(\varphi(\varepsilon_0,0)\) \([[[0_2]_2]_2]\) has level \(\vartheta(\Omega)\) \([[1[0_2]_2]_2]\) has level \(\vartheta(\Omega+1)\) \([[0_2[0_2]_2]_2]\) has level \(\vartheta(\Omega2)\) \([[1_2[0_2]_2]_2]\) has level \(\vartheta(\Omega\omega)\) \([[[0_2]_2[0_2]_2]_2]\) has level \(\vartheta(\Omega^2)\) \([[[10_2]_2]_2]\) has level \(\vartheta(\Omega^\omega)\) \([[[0_20_2]_2]_2]\) has level \(\vartheta(\Omega^\Omega)\) \([[[1_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega\omega})\) \([[[[0_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^2})\) \([[1[[0_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^2}+1)\) \([[[0_2[0_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^2+\Omega})\) \([[[1_2[0_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^2+\Omega\omega})\) \([[[[10_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^2\omega})\) \([[[[0_20_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^3})\) \([[[[1_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^\omega})\) \([[[[[0_2]_2]_2]_2]_2]\) has level \(\vartheta(\Omega^{\Omega^\Omega})\) \([0_3]\) has level \(\vartheta(\varepsilon_{\Omega+1})\) Linear Array Notation Here are no $, but this are just rules what you should do if that kind of array is the lowest level array. 7. \(b\bullet,c = [0,c-1_{b-1\bullet,c1}]\) 8. To diagonalize in the nth position with bracket types, you must use \(\underbrace{0,0...0,1}_n_k\) They diagonalize in the last entry, for any type of bracket. 9. \(\diamond,b\bullet,c,\bullet = [[\diamond,\diamond,b\bullet,c-1,\bullet_{\diamond,b-1\bullet,c,\bullet},c-1,\bullet]\) 10. \(0,c,\bullet = 0\) S3. Zeroes at the and of the array must be removed Analysis \([[0,1]]\) has level \(\psi(\psi_I(0))\) \([[00,1]]\) has level \(\psi(\psi_I(1))\) \([[1,1]]\) has level \(\psi(\psi_I(\omega))\) \([[[0_2],1]]\) has level \(\psi(\psi_I(\varepsilon_0))\) \([[[[0_2]_2],1]]\) has level \(\psi(\psi_I(\zeta_0))\) \([[[[0_3]_2],1]]\) has level \(\psi(\psi_I(\varphi(\omega,0)))\) \([[[[0,1]],1]]\) has level \(\psi(\psi_I(\psi(\psi_I(0))))\) \([[[[[[0,1]],1]],1]]\) has level \(\psi(\psi_I(\psi(\psi_I(\psi(\psi_I(0))))))\) \([[0_2,1]]\) has level \(\psi(\psi_I(\Omega))\) \([[0_{0},1]]\) has level \(\psi(\psi_I(\Omega_\omega))\) \([[[0,1],1]]\) has level \(\psi(\psi_I(\psi_I(0)))\) \([[0,1_2,1]]\) has level \(\psi(\psi_I(I))\) \([[1,1_2,1]]\) has level \(\psi(\psi_I(I\omega))\) \([[[0_2,1]_2,1]]\) has level \(\psi(\psi_I(I\Omega))\) \([[[0,1,1]_2,1]]\) has level \(\psi(\psi_I(I\psi_I(0)))\) \([[[0,1_2,1]_2,1]]\) has level \(\psi(\psi_I(I^2))\) \([[[1,1_2,1]_2,1]]\) has level \(\psi(\psi_I(I^\omega))\) \([[[[0,1_2,1]_2,1]_2,1]]\) has level \(\psi(\psi_I(I^I))\) \([[0,1_3,1]]\) has level \(\psi(\psi_I(\varepsilon_{I+1}))\) \([[[[0,1]_3,1]_3,1]]\) has level \(\psi(\psi_I(\varphi(\omega,I+1)))\) \([[[[0,1_3,1]_3,1]_3,1]]\) has level \(\psi(\psi_I(\Omega_{I+1}))\) \([[0,2]]\) has level \(\psi(\psi_{I_2}(0))\) \([[0,0]]\) has level \(\psi(\psi_{I_\omega}(0))\) \([[0,0_2]]\) has level \(\psi(\psi_{I_\Omega}(0))\) \([[0,0,1]]\) has level \(\psi(\psi_{I_{\psi_I(0)}}(0))\) \([[0,0,2]]\) has level \(\psi(\psi_{I_{\psi_{I_2}(0)}}(0))\) \([[0,0,1_2]]\) has level \(\psi(\psi_{I_{I}}(0))\) \([[0,0,1]]\) has level \(\psi(\psi_{\chi(1)}(0))\) \([[0,00,1]]\) has level \(\psi(\psi_{\chi(1)}(1))\) \([[0,0_2,1]]\) has level \(\psi(\psi_{\chi(1)}(\Omega))\) \([[0,0,0,1_2,1]]\) has level \(\psi(\psi_{\chi(1)}(\chi(1)))\) \(0,0,2\) has level \(\psi(\psi_{\chi(2)}(0))\) \(0,0,3\) has level \(\psi(\psi_{\chi(3)}(0))\) \(0,0,[0]\) has level \(\psi(\psi_{\chi(\omega)}(0))\) \([[0,0,0_2]]\) has level \(\psi(\psi_{\chi(\Omega)}(0))\) \(0,0,[0,1]\) has level \(\psi(\psi_{\chi(\psi_I(0))}(0))\) \([[0,0,0,0,1_2]]\) has level \(\psi(\psi_{\chi(M)}(0))\) \([[0,0,0,1]]\) has level \(\psi(\Psi_{\Xi(3,0)}(0))\) \([[0,0,0,0,1]]\) has level \(\psi(\Psi_{\Xi(4,0)}(0))\) limit of linear arrays is \(\psi(\Psi_{\Xi(\omega,0)}(0))\) Extended arrays Entry counter 11. \(0e^c0 = 0\) 12. \(0e(b\bullet) = 0,0...0,1e(b-1\bullet)\) a entries limit of the entry counter is \(\psi(\Psi_{\Xi(K)}(0))\) Dimensional arrays&Extended entry counter 11. \(0e^c0 = 0\) 12. \(0e^c(b\bullet) = 0(c)0...0(c)1e^c(b-1\bullet)\) a entries 13. \(\diamond(c)b = \diamond(c)b-1e^c(\diamond(c)b-1e^c(\diamond(c)b-1e^c(...)))\) where the \(e^c\) operator works on the first dimension before © and there are no lower dimensions in \(\diamond\) An comma is can be written shorthand for (0). limit of dimensional arrays is \(S(T^\omega)\). Nested Array Notation 14. \(b\bullet(\diamond,0,c)1 = [0(\diamond,b-1\bullet(\diamond,0,c)1b-1\bullet(\diamond,0,c)1,c-1)1]\) limit of nested arrays is \(\varepsilon_0\) ordinal structure. The S function In the first row, we have \(\Omega\), \(I\), \(M\),\(\Xi(3,0)\), ... In the second row, we have \(K\), \(\Xi_2(1,0)\), \(\Xi_2(2,0)\), ... It is inaccessible cardinal diagonalizing over \(K\) instead of \(\Omega\) After that, we can also have a third row, a forth row, a plane, up to n dimensions. Then we can have a diagonalizer over the dimensions, which can be a new dimension. Of course be can extend it to rows of dimensions, n-dimensions of dimensions, n-dimensions of dimensions of dimensions, ... A function from Hyp cos: \(S(\alpha,\beta,\gamma,\delta,...)\) = The \(\alpha\)th ordinal in the \(\beta\)th row in the \(\gamma\)th plane in the \(\delta\)th 3-dimension... This is like the Velben hierarchy, so you can extend it with \(S(T^\omega)\), \(S(T^T)\), \(S(\varepsilon_{T+1})\) Category:Blog posts